Complexity of local search for the p-median problem

We study the complexity of finding local minima for the p-median problem. The relationship between Swap local optima, 0–1 local saddle points, and classical Karush–Kuhn–Tucker conditions is presented. It is shown that the local search problems with some neighborhoods are tight PLS-complete. Moreover, the standard local descent algorithm takes exponential number of iterations in the worst case regardless of the tie-breaking and pivoting rules used. To illustrate this property, we present a family of instances where some local minima may be hard to find. Computational results with different pivoting rules for random and Euclidean test instances are discussed. These empirical results show that the standard local descent algorithm is polynomial in average for some pivoting rules.     

Hybrid and multi-field variational principles for geometrically exact three-dimensional beams

This paper addresses the development of several alternative novel hybrid/multi-field variational formulations of the geometrically exact three-dimensional elastostatic beam boundary-value problem. In the framework of the complementary energy-based formulations, a Legendre transformation is used to introduce the complementary energy density in the variational statements as a function of stresses only. The corresponding variational principles are shown to feature stationarity within the framework of the boundary-value problem. Both weak and linearized weak forms of the principles are presented. The main features of the principles are highlighted, giving special emphasis to their relationships from both theoretical and computational standpoints.     

三角模糊数互补判断矩阵的一种排序方法

给出三角模糊数互补判断矩阵的概念及三角模糊数相互比较的可能度公式 ,提出一种基于可能度的三角模糊数互补判断矩阵排序方法 ,通过算例说明该方法的可行性和有效性。     

Solving quadratic convex bilevel programming problems using a smoothing method

In this paper, we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem. We use the Karush-Kuhn-Tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints; the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty. These complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using Fisher-Burmeister functional. Then, using a quadratic sequential programming method, we solve a series of smooth, regular problems that progressively approximate the nonsmooth problem. Some preliminary computational results are reported, showing that our approach is efficient.     

Increasing data locality and introducing Level-3 BLAS in the Neville elimination

In this paper we present two new algorithmic variants to compute the Neville elimination, with and without pivoting, which improve data locality and cast most of the computations in terms of high-performance Level 3 BLAS. The experimental evaluation on a state-of-the-art multi-core processor demonstrates that the new blocked algorithms exhibit a much higher degree of concurrency and better cache usage, yielding higher performance while offering numerical accuracy akin to that of the traditional columnwise variant in most cases.     

弹性大变形问题中的应力状态描述、奇异性问题和余能原理

对弹性大变形理论中的3方面问题进行了综述.首先,对各种应变度量的共轭应力进行综述.大变形问题引起的应力状态描述的复杂性引起了许多学者的兴趣,对这个问题的研究也促进了大变形弹性理论的发展.在各种特定问题中,人们提出了不同的应力张量来描述应力状态,如Caucby应力张量、第一类和第一二类Piola-Kirchhoff应力张...     

Graph polynomials from principal pivoting

The recursive computation of the interlace polynomial introduced by Arratia, Bollobás and Sorkin is defined in terms of a new pivoting operation on undirected simple graphs. In this paper, we interpret the new pivoting operation on graphs in terms of standard pivoting (on matrices). Specifically, we show that, up to swapping vertex labels, Arratia et al.'s pivoting operation on a graph is equivalent to a principal pivot transform on the graph's adjacency matrix, provided that all computations are performed in the Galois field F₂. Principal pivoting on adjacency matrices over F₂ has a natural counterpart on isotropic systems. Thus, our view of the interlace polynomial is closely related to the one by Aigner and van der Holst.The observations that adjacency matrices of undirected simple graphs are skew-symmetric in F₂ and that principal pivoting preserves skew-symmetry in all fields suggest to extend Arratia et al.'s pivoting operation to fields other than F₂. Thus, the interlace polynomial extends to polynomials on gain graphs, namely bidirected edge-weighted graphs whereby reversed edges carry non-zero weights that differ only by their sign. Extending a proof by Aigner and van der Holst, we show that the extended interlace polynomial can be represented in a non-recursive form analogous to the non-recursive form of the original interlace polynomial, i.e., the Martin polynomial.For infinite fields it is shown that the extended interlace polynomial does not depend on the (non-zero) gains, as long as they obey a non-singularity condition. These gain graphs are all supported by a single undirected simple graph. Thus, a new graph polynomial is defined for undirected simple graphs. The recursive computation of the new polynomial can be done such that all ends of the recursion correspond to independent sets. Moreover, its degree equals the independence number. However, the new graph polynomial is different from the independence polynomial.     

Multifield hyperelasticity: variational theorems for complex bodies

We present some multifield variational principles for complex elastic solids characterized by any material microstructure. Some general constitutive arguments in the context of multifield hyperelasticity are also discussed.     

The complementary unboundedness of dual feasible solution sets in convex programming

F.E. Clark has shown that if at least one of the feasible solution sets for a pair of dual linear programming problems is nonempty then at least one of them is both nonempty and unbounded. Subsequently, M. Avriel and A.C. Williams have obtained the same result in the more general context of (prototype posynomial) geometric programming. In this paper we show that the same result is actually false in the even more general context of convex programming — unless a certain regularity condition is satisfied.We also show that the regularity condition is so weak that it is automatically satisfied in linear programming (prototype posynomial) geometric programming, quadratic programming (with either linear or quadratic constraints),l _p -regression analysis, optimal location, roadway network analysis, and chemical equilibrium analysis. Moreover, we develop an equivalent regularity condition for each of the usual formulations of duality.Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-73-2516.     

Polyhedral extensions of some theorems of linear programming

Three theorems of linear programming form our starting point: Tucker's theorem (1956) concerning the existence of optimal solutions satisfying the complementary slackness conditions strictly, and Williams' two theorems (1970) concerning the coordinatewise complementary behavior of feasible and optimal solutions. Here, we establish that the same phenomena hold in another, more versatile framework involving general polyhedral convexity. As one main application, the results are transferred into the context of the monotone complementarity problem. Several other theoretical applications are indicated.Research supported in part by the National Science Foundation under grant number MCS 79-05793 at the University of Illinois at Urbana-Champaign.